Independent solution

How to solve this Exponential Distribution question

Setup

Setup

Use the median condition to determine the exponential rate.

Pr⁡(T>10)=e−10c=0.5\Pr(T>10)=e^{-10c}=0.5
c=ln⁡210c=\frac{\ln2}{10}

Model

Model

Separate the two payment levels into disjoint time intervals.

E[W]=35Pr⁡(T≤5)+25Pr⁡(5<T≤7.5)E[W]=35\Pr(T\le5)+25\Pr(5<T\le7.5)
F(t)=1−e−ctF(t)=1-e^{-ct}

Compute

Compute

Evaluate the exponential probabilities at both warranty boundaries.

E[W]=35F(5)+25(F(7.5)−F(5))E[W]=35F(5)+25\bigl(F(7.5)-F(5)\bigr)
E[W]=35(1−2−1/2)+25(2−1/2−2−3/4)E[W]=35(1-2^{-1/2})+25(2^{-1/2}-2^{-3/4})
E[W]=13.0638432506…E[W]=13.0638432506\ldots

Answer

Answer

The expected warranty payment rounds to 13.06.

13.06(A)\boxed{13.06\quad\text{(A)}}